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**Chapter 8 Introduction to Hypothesis Testing**

PowerPoint Lecture Slides Essentials of Statistics for the Behavioral Sciences Eighth Edition by Frederick J. Gravetter and Larry B. Wallnau

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**Chapter 8 Learning Outcomes**

1 Understand logic of hypothesis testing 2 State hypotheses and locate critical region(s) 3 Conduct z-test and make decision 4 Define and differentiate Type I and Type II errors 5 Understand effect size and compute Cohen’s d 6 Make directional hypotheses and conduct one-tailed test

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**Tools You Will Need z-Scores (Chapter 5)**

Distribution of sample means (Chapter 7) Expected value Standard error Probability and sample means

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**8.1 Hypothesis Testing Logic**

Hypothesis testing is one of the most commonly used inferential procedures Definition: a statistical method that uses sample data to evaluate the validity of a hypothesis about a population parameter

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**Logic of Hypothesis Test**

State hypothesis about a population Predict the expected characteristics of the sample based on the hypothesis Obtain a random sample from the population Compare the obtained sample data with the prediction made from the hypothesis If consistent, hypothesis is reasonable If discrepant, hypothesis is rejected

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**Figure 8.1 Basic Experimental Design**

FIGURE 8.1 The basic experimental situation for hypothesis testing. It is assumed that the parameter μ is known for the population before treatment. The purpose of the experiment is to determine whether the treatment has an effect on the population mean.

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**Figure 8.2 Unknown Population in Basic Experimental Design**

FIGURE 8.2 From the point of view of the hypothesis test, the entire population receives the treatment and then a sample is selected from the treated population. In the actual research study, a sample is selected from the original population and the treatment is administered to the sample. From either perspective, the result is a treated sample that represents the treated population.

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**Four Steps in Hypothesis Testing**

Step 1: State the hypotheses Step 2: Set the criteria for a decision Step 3: Collect data; compute sample statistics Step 4: Make a decision

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**Step 1: State Hypotheses**

Null hypothesis (H0) states that, in the general population, there is no change, no difference, or is no relationship Alternative hypothesis (H1) states that there is a change, a difference, or there is a relationship in the general population

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**Step 2: Set the Decision Criterion**

Distribution of sample outcomes is divided Those likely if H0 is true Those “very unlikely” if H0 is true Alpha level, or significance level, is a probability value used to define “very unlikely” outcomes Critical region(s) consist of the extreme sample outcomes that are “very unlikely” Boundaries of critical region(s) are determined by the probability set by the alpha level

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**Figure 8.3 Note “Unlikely” Parts of Distribution of Sample Means**

FIGURE 8.3 The set of potential samples is divided into those that are likely to be obtained and those that are very unlikely to be obtained if the null hypothesis is true.

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**Figure 8.4 Critical region(s) for α = .05**

FIGURE 8.4 The critical region (very unlikely outcomes) for alpha = .05

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Learning Check A sports coach is investigating the impact of a new training method. In words, what would the null hypothesis say? A The new training program produces different results from the existing one B The new training program produces results about like the existing one C The new training program produces better results than the existing one D There is no way to predict the results of the new training program

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**Learning Check - Answer**

A sports coach is investigating the impact of a new training method. In words, what would the null hypothesis say? A The new training program produces different results from the existing one B The new training program produces results about like the existing one C The new training program produces better results than the existing one D There is no way to predict the results of the new training program

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Learning Check Decide if each of the following statements is True or False. T/F If the alpha level is decreased, the size of the critical region decreases The critical region defines unlikely values if the null hypothesis is true

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**Learning Check - Answers**

True Alpha is the proportion of the area in the critical region(s) This is the definition of “unlikely”

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**Step 3: Collect Data (and…)**

Data always collected after hypotheses stated Data always collected after establishing decision criteria This sequence assures objectivity

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**Step 3: (continued)… Compute Sample Statistics**

Compute a sample statistic (z-score) to show the exact position of the sample In words, z is the difference between the observed sample mean and the hypothesized population mean divided by the standard error of the mean

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Step 4: Make a decision If sample statistic (z) is located in the critical region, the null hypothesis is rejected If the sample statistic (z) is not located in the critical region, the researcher fails to reject the null hypothesis

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**Jury Trial: Hypothesis Testing Analogy**

Trial begins with the null hypothesis “not guilty” (defendant’s innocent plea) Police and prosecutor gather evidence (data) relevant to the validity of the innocent plea With sufficient evidence against, jury rejects null hypothesis innocence claim to conclude “guilty” With insufficient evidence against, jury fails to convict, i.e., fails to reject the “not guilty” claim (but does not conclude defendant is innocent)

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Learning Check Decide if each of the following statements is True or False. T/F When the z-score is quite extreme, it shows the null hypothesis is true A decision to retain the null hypothesis means you proved that the treatment has no effect

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**Learning Check - Answer**

False An extreme z-score is in the critical region—very unlikely if H0 is true Failing to reject H0 does not prove it true; there is just not enough evidence to reject it

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**8.2 Uncertainty and Errors in Hypothesis Testing**

Hypothesis testing is an inferential process Uses limited information from a sample to make a statistical decision, and then from it a general conclusion Sample data used to make the statistical decision allows us to make an inference and draw a conclusion about a population Errors are possible

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Type I Errors Researcher rejects a null hypothesis that is actually true Researcher concludes that a treatment has an effect when it has none Alpha level is the probability that a test will lead to a Type I error

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Type II Errors Researcher fails to reject a null hypothesis that is really false Researcher has failed to detect a real treatment effect Type II error probability is not easily identified

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**Table 8.1 Type I error (α) Type II error (β) Actual Situation**

No Effect = H0 True Effect Exists = H0 False Researcher’s Decision Reject H0 Type I error (α) Decision correct Fail to reject H0 Type II error (β)

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**Figure 8.5 Location of Critical Region Boundaries**

FIGURE 8.5 The locations of the critical region boundaries for three different levels of significance: α = .05, α = .01, and α = .001.

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Learning Check Decide if each of the following statements is True or False. T/F A Type I error is like convicting an innocent person in a jury trial A Type II error is like convicting a guilty person in a jury trial

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**Learning Check - Answer**

True Innocence is the “null hypothesis” for a jury trial; conviction is like rejecting that hypothesis False Convicting a guilty person is not an error; but acquitting a guilty person would be like Type II error

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**8.3 Hypothesis Testing Summary**

Step 1: State hypotheses and select alpha level Step 2: Locate the critical region Step 3: Collect data; compute the test statistic Step 4: Make a probability-based decision about H0: Reject H0 if the test statistic is unlikely when H0 is true—called a “significant” or “statistically significant” result

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In the Literature A result is significant or statistically significant if it is very unlikely to occur when the null hypothesis is true; conclusion: reject H0 In APA format Report that you found a significant effect Report value of test statistic Report the p-value of your test statistic

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**Figure 8.6 Critical Region for Standard Test**

FIGURE 8.6 Sample means that fall in the critical region (shaded areas) have a probability less than alpha (p < α). In this case, H0 should be rejected. Sample means that do fall in the critical region have a probability greater than alpha (p > α).

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**8.3 Assumptions for Hypothesis Tests with z-Scores**

Random sampling Independent Observation Value of σ is not changed by the treatment Normally distributed sampling distribution

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**Factors that Influence the Outcome of a Hypothesis Test**

Size of difference between sample mean and original population mean Larger discrepancies larger z-scores Variability of the scores More variability larger standard error Number of scores in the sample Larger n smaller standard error

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**Learning Check σ = 5 and n = 25 σ = 5 and n = 50 σ = 10 and n = 25**

A researcher uses a hypothesis test to evaluate H0: µ = 80. Which combination of factors is most likely to result in rejecting the null hypothesis? A σ = 5 and n = 25 B σ = 5 and n = 50 C σ = 10 and n = 25 D σ = 10 and n = 50

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**Learning Check - Answer**

A researcher uses a hypothesis test to evaluate H0: µ = 80. Which combination of factors is most likely to result in rejecting the null hypothesis? A σ = 5 and n = 25 B σ = 5 and n = 50 C σ = 10 and n = 25 D σ = 10 and n = 50

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Learning Check Decide if each of the following statements is True or False. T/F An effect that exists is more likely to be detected if n is large An effect that exists is less likely to be detected if σ is large

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**Learning Check - Answers**

True A larger sample produces a smaller standard error and larger z A larger standard deviation increases the standard error and produces a smaller z

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**8.4 Directional Hypothesis Tests**

The standard hypothesis testing procedure is called a two-tailed (non-directional) test because the critical region involves both tails to determine if the treatment increases or decreases the target behavior However, sometimes the researcher has a specific prediction about the direction of the treatment

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**8.4 Directional Hypothesis Tests (Continued)**

When a specific direction of the treatment effect can be predicted, it can be incorporated into the hypotheses In a directional (one-tailed) hypothesis test, the researcher specifies either an increase or a decrease in the population mean as a consequence of the treatment

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**Figure 8.7 Example 8.3 Critical Region (Directional)**

FIGURE 8.7 Critical region for Example 8.3.

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**One-tailed and Two-tailed Tests Compared**

One-tailed test allows rejecting H0 with relatively small difference provided the difference is in the predicted direction Two-tailed test requires relatively large difference regardless of the direction of the difference In general two-tailed tests should be used unless there is a strong justification for a directional prediction

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Learning Check A researcher is predicting that a treatment will decrease scores. If this treatment is evaluated using a directional hypothesis test, then the critical region for the test. A would be entirely in the right-hand tail of the distribution B would be entirely in the left-hand tail of the distribution C would be divided equally between the two tails of the distribution D cannot answer without knowing the value of the alpha level

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**Learning Check - Answer**

A researcher is predicting that a treatment will decrease scores. If this treatment is evaluated using a directional hypothesis test, then the critical region for the test. A would be entirely in the right-hand tail of the distribution B would be entirely in the left-hand tail of the distribution C would be divided equally between the two tails of the distribution D cannot answer without knowing the value of the alpha level

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**8.5 Hypothesis Testing Concerns: Measuring Effect Size**

Although commonly used, some researchers are concerned about hypothesis testing Focus of test is data, not hypothesis Significant effects are not always substantial Effect size measures the absolute magnitude of a treatment effect, independent of sample size Cohen’s d measures effect size simply and directly in a standardized way

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**Cohen’s d : Measure of Effect Size**

Magnitude of d Evaluation of Effect Size d = 0.2 Small effect d = 0.5 Medium effect d = 0.8 Large effect

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**Figure 8.8 When is a 15-point Difference a “Large” Effect?**

FIGURE 8.8 The appearance of a 15-point treatment effect in two different situations. In part (a), the standard deviation is σ = 100 and the 15-point effect is relatively small. In part (b), the standard deviation is σ = 15 and the 15-point effect is relatively large. Cohen’s d uses the standard deviation to help measure effect size.

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Learning Check Decide if each of the following statements is True or False. T/F Increasing the sample size will also increase the effect size Larger differences between the sample and population mean increase effect size

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**Learning Check -Answers**

False Sample size does not affect Cohen’s d True The mean difference is in the numerator of Cohen’s d

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8.6 Statistical Power The power of a test is the probability that the test will correctly reject a false null hypothesis It will detect a treatment effect if one exists Power = 1 – β [where β = probability of a Type II error] Power usually estimated before starting study Requires several assumptions about factors that influence power

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**Figure 8.9 Measuring Statistical Power**

FIGURE 8.9 A demonstration of measuring power for a hypothesis test. The left-hand side shows the distribution of sample means that would occur if the null hypothesis is true. The critical region is defined for this distribution. The right-hand side shows the distribution of sample means that would be obtained if there were an 8-point treatment effect. Notice that, if there is an 8-point effect, essentially all of the sample means would be in the critical region. Thus, the probability of rejecting H0 (the power of the test) would be nearly 100% for an 8-point treatment effect.

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**Influences on Power Increased Power Decreased Power**

As effect size increases, power also increases Larger sample sizes produce greater power Using a one-tailed (directional) test increases power (relative to a two-tailed test) Decreased Power Reducing the alpha level (making the test more stringent) reduces power Using two-tailed (non-directional) test decreases power (relative to a one-tailed test)

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**Figure 8.10 Sample Size Affects Power**

FIGURE A demonstration of how sample size affects the power of a hypothesis test. As in Figure 8.9, the left-hand side shows the distribution of sample means if the null hypothesis were true. The critical region is defined for this distribution. The right-hand side shows the distribution of sample means that would be obtained if there were an 8-point treatment effect. Notice that reducing the sample size to n = 4 has reduced the power of the test to less than 50% compared to a power of nearly 100% with a sample of n = 25 in Figure 8.9.

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Learning Check The power of a statistical test is the probability of _____ A rejecting a true null hypothesis B supporting true null hypothesis C rejecting a false null hypothesis D supporting a false null hypothesis

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**Learning Check - Answer**

The power of a statistical test is the probability of _____ A rejecting a true null hypothesis B supporting true null hypothesis C rejecting a false null hypothesis D supporting a false null hypothesis

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Learning Check Decide if each of the following statements is True or False. T/F Cohen’s d is used because alone, a hypothesis test does not measure the size of the treatment effect Lowering the alpha level from .05 to .01 will increase the power of a statistical test

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**Answer Differences might be significant but not of substantial size**

True Differences might be significant but not of substantial size False It is less likely that H0 will be rejected with a small alpha

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Any Questions? Concepts? Equations?

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